6 ms·

Generalizations of Fourier analysis (2021)

130 points by mscharrer 3y ago

lixtra 3y ago
This sounds like a great direction for an advanced seminar in university.
macrolocal 3y ago
Yosida is a great reference for this functional analysis. For a much broader generalization, albeit with expensive concepts, cf. Tannaka-Krein duality.
mananaysiempre 3y ago
A couple of other things that AFAIK aren’t special cases of the ones in the list: - The idempotent (“tropical”) Fourier transform turns out to be the Legendre transform; - The fractional Fourier transform, known to physicists as the propagator of the quantum harmonic oscillator, is a pretty fun thing to consider; - The Fourier-Laplace transform on Abelian groups seems like a fairly straightforward extension of the idea of plugging in a complex frequency, but I haven’t seen a textbook exposition (only an old article); - The non-linear Fourier transform (with Ki xi + Lij xi xj + ..., finite or infinite sum) seems impressively obscure (I know of a total of one book reference) but occurs in quantum field theory as the “n-loop” or “∞-loop effective action”; - The odd (in the super sense) Fourier transform turns out to underpin stuff like the Hodge star on differential forms; - On a finite non-Abelian groups, the duality splits into two: every function on conjugacy classes is a linear combination of irreducible characters; every function on group is a linear combination of irreducible matrix elements; this is probably also doable on Lie groups but I’m too much of a wimp to learn the theory. (Also, generating functions should by all appearances be a fairly elementary chapter of the Fourier story, as electronic engineers with their “Z-transform” also realize, but I haven’t seen that implemented convincingly in full.) See as well Baez’s old issue of “This Week’s Finds” where he started with sound and well all the way to spectra of Banach algebras and rings—as in Gelfand duality, algebraic geometry etc. (Can’t seem to locate the specific issue now.) Of course there are also wavelets (there’s even a Fields Medal for those now), but I don’t know that they fit into the representation theory ideology (would be excited to be wrong!).
- dustingetz 3y ago
  what is an intuition for complex frequency?
  - paulsutter 3y ago
    Signals estimated by the FFT have two parameters: magnitude and phase. FFT results evade intuition because complex numbers are cartesian. If you convert them to polar coordinates they make more sense as magnitude and phase https://www.gaussianwaves.com/2015/11/interpreting-fft-results-obtaining-magnitude-and-phase-information/ https://www.gaussianwaves.com/2015/11/interpreting-fft-resul... Note that complex numbers are merely convenient for working with two dimensional quantities. The square root of -1 is just math geek for orthogonal, and has nothing whatsoever to do with signals
    - mananaysiempre 3y ago
      (Note: GGP, not GP.) I meant complex frequency and not complex amplitude though.
      - jdhwosnhw 3y ago
        From a physics perspective, complex frequency results in “evanescent waves” - ie, waves that decay rather than oscillate (technically a fully complex frequency of the form a+ib will both oscillate and decay)
        nextaccountic 3y ago
        Yep: e^iω (with ω real) is an oscillation, but e^σ (with σ real) is an exponential decay when σ < 1, an exponential growth with σ > 0 and constant with σ = 1 so e^(σ + iω) = e^σ * e^iω is just an exponential growth or decay modulated by a sinusoid.. or, if σ is one, is just a pure oscillation ω is the usual frequency, but σ + iω is the complex frequency. the fourier transform deals with function that receives ω as input, and the laplace transform deals with functions that receives σ + iω instead. so the fourier transform is just a special case of the laplace transform with σ = 0
antognini 3y ago
I am reminded somewhat of a line in Sanjeev Arora's lecture notes A Theorist's Toolkit: "Sanjeev admits that he used to find Fourier transforms intimidating as a student. His fear vanished once he realized that it is a rewording of the following trivial idea: If u_1, u_2, ..., u_n is an orthonormal basis of R^n then every vector v can be expressed as Sum_i alpha_i u_i where alpha_i = <v, u_i> and Sum_i alpha_i^2 = |v|^2" https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf
- ajkjk 3y ago
  That's not a good enough statement to really summarize Fourier transforms, I think? It really just summarizes the idea of an orthogonal basis.
rthomas6 3y ago
Ugh, I need to learn more math. How do I even start? I know multi variable calculus, I know the basics of linear algebra, and I know Fourier transforms. Yet this article is half gibberish to me.
- adgjlsfhk1 3y ago
  imo abstract algebra is pretty much the gateway to most modern math. it well feel at first like it's a lot of machinery without purpose, but understanding groups rings and fields well opens the door to topology, advanced number theory and a bunch of the rest of math. the other option would be to learn some real and complex analysis, but imo the algebra side is where a lot more of the cool stuff is.
- nhatcher 3y ago
  I wish I could help you. If you are any sort of programmer "Linear Algebra" is going to be the best bang for your buck. You can have a look a some of the books of Gilbert Strang or his online courses. Beyond that the classic books in "Abstract Algebra" are those by Serge Lang or Jacobson. Either or might be too difficult and maybe not worth it. I always tell people to study integration correctly. By that I mean measure theory, with the Riemann integral taught in schools you can only go so far. With those two you are in a good position to tackle functional analysis. The classic book on the subject is that of Walter Rudin but maybe it too hard for self learners. A better alternative might be the open course at MIT: https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ https://ocw.mit.edu/courses/18-102-introduction-to-functiona... I also like the book by Vladimir Kadets. Functional analysis is terribly important in practical areas like signal processing, quantum mechanics or partial differential equation in general. so it might be worth your pain. There are many ways you can go from there. I can't leave without recommending one more things. If you are curious of advanced mathematics as a tourist the three books by Avner Ash and Robert Gross make a great read
superb-owl 3y ago
> Notice that, even if their formulas look quite similar, the Fourier series is not a particular case of the Fourier transform. For example, a periodic function is never integrable over the real line unless it is identically zero. Thus, you cannot compute the Fourier transform of a periodic function. Someone correct me if I'm wrong, but I do think the latter does generalize the former. I vaguely remember seeing it derived as essentially linking +/- infinity so the function is "periodic" on the real line. But I could be misremembering This is the class I took, it's incredible: https://see.stanford.edu/Course/EE261/137 https://see.stanford.edu/Course/EE261/137
Paul-Craft 3y ago
Let's not forget the graph Fourier transform: https://en.wikipedia.org/wiki/Graph_Fourier_transform https://en.wikipedia.org/wiki/Graph_Fourier_transform
- enriquto 3y ago
  Yes. It is the discrete analogue of spectral geometry on manifolds. A graph is a discrete manifold.
StarlaAtNight 3y ago
I asked ChatGPT to explain the intro section of this article like I'm five: """ Fourier analysis is like playing with music notes. We have different ways of looking at these notes, like how high or low they are, or how long they last. In the same way, we have different ways of looking at signals and how they change over time or space. Just like how we can make different types of music by combining different notes in different ways, we can make different types of signals by combining different types of Fourier analysis. There are four classic ways to do Fourier analysis, and each one is like a different way of looking at the signal. They're like different types of glasses we can put on to see the signal more clearly. But there are also other ways to do Fourier analysis, like looking at signals on different types of spaces or using different types of math. It's like having even more types of glasses to look at the signal! So when you ask a mathematician what Fourier analysis is, they might give you a different answer depending on which type of glasses they like to wear. """

130 points by mscharrer 3y ago

This sounds like a great direction for an advanced seminar in university.

Yosida is a great reference for this functional analysis. For a much broader generalization, albeit with expensive concepts, cf. Tannaka-Krein duality.

mananaysiempre 3y ago

A couple of other things that AFAIK aren’t special cases of the ones in the list: - The idempotent (“tropical”) Fourier transform turns out to be the Legendre transform; - The fractional Fourier transform, known to physicists as the propagator of the quantum harmonic oscillator, is a pretty fun thing to consider; - The Fourier-Laplace transform on Abelian groups seems like a fairly straightforward extension of the idea of plugging in a complex frequency, but I haven’t seen a textbook exposition (only an old article); - The non-linear Fourier transform (with Ki xi + Lij xi xj + ..., finite or infinite sum) seems impressively obscure (I know of a total of one book reference) but occurs in quantum field theory as the “n-loop” or “∞-loop effective action”; - The odd (in the super sense) Fourier transform turns out to underpin stuff like the Hodge star on differential forms; - On a finite non-Abelian groups, the duality splits into two: every function on conjugacy classes is a linear combination of irreducible characters; every function on group is a linear combination of irreducible matrix elements; this is probably also doable on Lie groups but I’m too much of a wimp to learn the theory. (Also, generating functions should by all appearances be a fairly elementary chapter of the Fourier story, as electronic engineers with their “Z-transform” also realize, but I haven’t seen that implemented convincingly in full.) See as well Baez’s old issue of “This Week’s Finds” where he started with sound and well all the way to spectra of Banach algebras and rings—as in Gelfand duality, algebraic geometry etc. (Can’t seem to locate the specific issue now.) Of course there are also wavelets (there’s even a Fields Medal for those now), but I don’t know that they fit into the representation theory ideology (would be excited to be wrong!).

dustingetz 3y ago
what is an intuition for complex frequency?
- paulsutter 3y ago
  Signals estimated by the FFT have two parameters: magnitude and phase. FFT results evade intuition because complex numbers are cartesian. If you convert them to polar coordinates they make more sense as magnitude and phase https://www.gaussianwaves.com/2015/11/interpreting-fft-results-obtaining-magnitude-and-phase-information/ https://www.gaussianwaves.com/2015/11/interpreting-fft-resul... Note that complex numbers are merely convenient for working with two dimensional quantities. The square root of -1 is just math geek for orthogonal, and has nothing whatsoever to do with signals
  - mananaysiempre 3y ago
    (Note: GGP, not GP.) I meant complex frequency and not complex amplitude though.
    - jdhwosnhw 3y ago
      From a physics perspective, complex frequency results in “evanescent waves” - ie, waves that decay rather than oscillate (technically a fully complex frequency of the form a+ib will both oscillate and decay)
      - nextaccountic 3y ago
        Yep: e^iω (with ω real) is an oscillation, but e^σ (with σ real) is an exponential decay when σ < 1, an exponential growth with σ > 0 and constant with σ = 1 so e^(σ + iω) = e^σ * e^iω is just an exponential growth or decay modulated by a sinusoid.. or, if σ is one, is just a pure oscillation ω is the usual frequency, but σ + iω is the complex frequency. the fourier transform deals with function that receives ω as input, and the laplace transform deals with functions that receives σ + iω instead. so the fourier transform is just a special case of the laplace transform with σ = 0

antognini 3y ago

I am reminded somewhat of a line in Sanjeev Arora's lecture notes A Theorist's Toolkit: "Sanjeev admits that he used to find Fourier transforms intimidating as a student. His fear vanished once he realized that it is a rewording of the following trivial idea: If u_1, u_2, ..., u_n is an orthonormal basis of R^n then every vector v can be expressed as Sum_i alpha_i u_i where alpha_i = <v, u_i> and Sum_i alpha_i^2 = |v|^2" https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf

ajkjk 3y ago
That's not a good enough statement to really summarize Fourier transforms, I think? It really just summarizes the idea of an orthogonal basis.

rthomas6 3y ago

Ugh, I need to learn more math. How do I even start? I know multi variable calculus, I know the basics of linear algebra, and I know Fourier transforms. Yet this article is half gibberish to me.

adgjlsfhk1 3y ago
imo abstract algebra is pretty much the gateway to most modern math. it well feel at first like it's a lot of machinery without purpose, but understanding groups rings and fields well opens the door to topology, advanced number theory and a bunch of the rest of math. the other option would be to learn some real and complex analysis, but imo the algebra side is where a lot more of the cool stuff is.
nhatcher 3y ago
I wish I could help you. If you are any sort of programmer "Linear Algebra" is going to be the best bang for your buck. You can have a look a some of the books of Gilbert Strang or his online courses. Beyond that the classic books in "Abstract Algebra" are those by Serge Lang or Jacobson. Either or might be too difficult and maybe not worth it. I always tell people to study integration correctly. By that I mean measure theory, with the Riemann integral taught in schools you can only go so far. With those two you are in a good position to tackle functional analysis. The classic book on the subject is that of Walter Rudin but maybe it too hard for self learners. A better alternative might be the open course at MIT: https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ https://ocw.mit.edu/courses/18-102-introduction-to-functiona... I also like the book by Vladimir Kadets. Functional analysis is terribly important in practical areas like signal processing, quantum mechanics or partial differential equation in general. so it might be worth your pain. There are many ways you can go from there. I can't leave without recommending one more things. If you are curious of advanced mathematics as a tourist the three books by Avner Ash and Robert Gross make a great read

superb-owl 3y ago

> Notice that, even if their formulas look quite similar, the Fourier series is not a particular case of the Fourier transform. For example, a periodic function is never integrable over the real line unless it is identically zero. Thus, you cannot compute the Fourier transform of a periodic function. Someone correct me if I'm wrong, but I do think the latter does generalize the former. I vaguely remember seeing it derived as essentially linking +/- infinity so the function is "periodic" on the real line. But I could be misremembering This is the class I took, it's incredible: https://see.stanford.edu/Course/EE261/137 https://see.stanford.edu/Course/EE261/137

Paul-Craft 3y ago

Let's not forget the graph Fourier transform: https://en.wikipedia.org/wiki/Graph_Fourier_transform https://en.wikipedia.org/wiki/Graph_Fourier_transform

enriquto 3y ago
Yes. It is the discrete analogue of spectral geometry on manifolds. A graph is a discrete manifold.

StarlaAtNight 3y ago

I asked ChatGPT to explain the intro section of this article like I'm five: """ Fourier analysis is like playing with music notes. We have different ways of looking at these notes, like how high or low they are, or how long they last. In the same way, we have different ways of looking at signals and how they change over time or space. Just like how we can make different types of music by combining different notes in different ways, we can make different types of signals by combining different types of Fourier analysis. There are four classic ways to do Fourier analysis, and each one is like a different way of looking at the signal. They're like different types of glasses we can put on to see the signal more clearly. But there are also other ways to do Fourier analysis, like looking at signals on different types of spaces or using different types of math. It's like having even more types of glasses to look at the signal! So when you ask a mathematician what Fourier analysis is, they might give you a different answer depending on which type of glasses they like to wear. """